029i - Translating Real Contexts to Algebra

Lesson 29 — Translating Real Contexts into Algebraic Expressions

Strand: Algebra | Descriptor: AC9M7A02 | Duration: 45 minutes

Learning Intentions

• To formulate algebraic expressions that model authentic situations.
• To interpret what each part of an expression represents in context.

Success Criteria

I can:

1. Choose and define a suitable variable for a situation.
2. Build an expression that captures the relationships described.
3. Explain what each term and coefficient means in context.
4. Use my expression to answer questions about the situation.

Warmup

(6 minutes — reverse translation, pairs)

For each expression, invent a situation it could describe. The variable is given.

1. — where is a number of items
2. — where is a number of hours
3. — where is a width in cm
4. — where is a total cost in dollars

Sample answers: 1. The cost of items at 125085$c6$ people.

Discussion: More than one situation fits each expression. What does that tell you about the relationship between algebra and the real world? (One expression can model many contexts — that generality is exactly what makes algebra powerful.)

Activities

Activity 1 — Explicit Instruction: the Modelling Protocol (12 min)

The four-step protocol:

1. Identify the quantity that varies — that becomes the variable.
2. Define it precisely, including units: “let be the number of tickets sold”.
3. Build the expression term by term, saying what each term contributes.
4. Check by substituting a sensible value and asking whether the result makes sense.

I do — the school fete. “A stall sells drinks for 3$45$ stall hire fee. Write an expression for the profit.”

Let be the number of drinks sold. Let profit be measured in dollars.

Narrate each part: is the money taken in; is the fixed cost, subtracted once.

Check: if , profit 15d = 10= 30 - 45 = -$15$ — a loss. Ask: what does a negative profit mean, and how many drinks are needed to break even? (15 drinks.)

We do — three contexts, built together:

1. A club has 500$35w$ weeks.
2. Tickets cost 18$11ac$ children.
3. A rectangle’s length is cm more than three times its width . Write an expression for its perimeter, simplified.

Activity 2 — Multi-variable and Multi-step Contexts (12 min)

Pairs. These require more than one variable or more than one step.

Problem 1 — Sports carnival. Entry costs 6$10$280$.

(a) Write an expression for the total cost with students and adults.

(b) Find the cost for students and adults.

(c) If the cost is shared equally among the students only, write an expression for each student’s share.

Problem 2 — Phone repair shop. A technician charges 60$45$p$.

(a) Write an expression for the total charge for a job of hours.

(b) Find the charge for a -hour job with 120$ of parts.

(c) What does the expression become if the call-out fee is waived for jobs over hours?

Problem 3 — Fencing. A rectangular paddock has width metres and length m longer than the width. Fencing costs 28$ per metre.

(a) Write an expression for the perimeter, simplified.

(b) Write an expression for the total fencing cost.

(c) Find the cost when m.

Socratic scaffolding for Problem 3:

Prompt	Purpose
Understand: what varies?	The width. Everything else follows from it.
Define the variable.	Let be the width in metres.
Express the length in terms of .	.
Write the perimeter before simplifying.	.
Expand and collect.	.
Now the cost. What does the perimeter represent here?	The total length of fencing needed.
Build the cost expression.	.
Carry it out at .	5040$.
Looking back — check another way	Perimeter m; 5040$ ✓

Answers: 1. (a) (b) 610\dfrac{6s + 10a + 280}{s}45h + p + 60135 + 120 + 60 = $31545h + p4w + 40112w + 1120$5040$.

Activity 3 — Inquiry: Interpreting Someone Else’s Expression (8 min)

Pairs. The reverse skill — reading meaning out of algebra.

A school canteen models its weekly profit with the expression

where is the number of rolls sold and the number of sandwiches.

1. What does the represent? The ? The ?
2. Find the profit when and .
3. In a quiet week the canteen sells rolls and sandwiches. What happens?
4. If no sandwiches are sold, how many rolls are needed to break even?

Socratic scaffolding for Q4:

Prompt	Purpose
Understand: what does “break even” mean?	Profit is exactly zero — no gain, no loss.
What happens to the expression if ?	It becomes .
Set up the condition.	.
Solve.	, so
Interpret in context.	You cannot sell part of a roll, and rolls still leaves a small loss — so 72 rolls are needed.
Looking back — check	, a loss of c. , a profit of 4$ ✓

Answers: 1. 4.50$2.80$320270 + 238 - 320 = $188180 + 84 - 320 = -$56$5672$ rolls.

Checks for Understanding

(5 minutes — exit ticket)

1. A shop sells shirts for 25$180n$ shirts.
2. Using your expression, find the profit from shirts.
3. A rectangle’s length is cm less than twice its width . Write a simplified expression for its perimeter.
4. Reasoning. In the expression modelling a job’s cost, explain what each number means.
5. A tank holds L and drains at L per hour. Write an expression for the volume after hours, and state when it is empty.

Answers: 1. ; 2. 120l = 2w - 4P = 2w + 2(2w-4) = 6w - 8$15$80600 - 25h25h = 60024$ hours.

Common Misconceptions

Misconception	How to pre-empt it
Not defining the variable, then losing track of what it means.	Make the definition line compulsory in every written answer.
Putting the fixed cost where the rate belongs.	Ask “is this charged once, or every time?” before writing each term.
Using the same letter for two different quantities.	Insist on distinct, meaningful letters, and list all definitions before building.
Forgetting to simplify, leaving as the final answer.	Require a simplified final line, and check it numerically against the unsimplified form.
Interpreting a negative result as an error rather than a loss.	Discuss explicitly what negative profit means, and use it to find break-even points.
Rounding a break-even figure the wrong way.	Ask “does this value actually satisfy the condition?” before rounding.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A market stall pays 75$4$200$?

Answer

So items are needed (68 gives 197$, just short).

E2 (AMC Junior style). Anna is years old. Ben is years older. Carla is twice Ben’s age. Write a simplified expression for the sum of all three ages, and find it when .

Answer

At : years total.

E3 (Challenge). A rectangular garden has width and length . A path m wide runs around the outside. Write a simplified expression for the area of the path.

Answer

E4 (Challenge). A hall hire costs 250$12$20nn$ giving a profit.

Answer

For : , so . The smallest whole number is guests (giving a profit of 6$).

E5 (Challenge). Three consecutive even numbers have the smallest equal to . Write a simplified expression for their sum, and show it is always a multiple of .

Answer

Since is even, is even, so is times an even number — a multiple of .

Homework

1. A cinema charges 16$9ac$ children.
2. A club has 800$60m9$ months. (c) After how many months is the money gone?
3. A rectangle’s length is cm more than twice its width . Write simplified expressions for (a) the length (b) the perimeter (c) the area.
4. A stall pays 120$7n40$ items. (c) How many items to break even?
5. A plumber charges 70$55$ph4$85$ of parts.
6. Fencing costs 32ss = 45$ m.
7. Reasoning. In modelling a stall’s profit, explain what each number means and what a negative value of would tell the stallholder.
8. Challenge. A rectangular pool of width and length is surrounded by a m paved border. Write a simplified expression for the area of the paving.

Answers: Q1 — . Q2 — (a) (b) 260800 - 60m = 0m = 13\tfrac13142w+86w+162w^2+8w7n - 120$160n = 17.14\ldots1855h + p + 70$375128s$5760$6$240P(w+4)(w+8) = w^2+12w+32w^2+4w= 8w + 32$.